A probabilistic version of a theorem of László Kovács and Hyo-Seob Sim

Document Type : Ischia Group Theory 2018

Authors

Dipartimento di Matematica Università di Padova

Abstract

For a finite group group‎, ‎denote by $\mathcal V(G)$ the smallest positive integer $k$ with the property that the probability of generating $G$ by $k$ randomly chosen elements is at least $1/e.$ Let $G$ be a finite soluble group‎. ‎{Assume} that for every $p\in \pi(G)$ there exists $G_p\leq G$ such that $p$ does not divide $|G:G_p|$ and ${\mathcal V}(G_p)\leq d.$ Then ${\mathcal V}(G)\leq d+7.$‎

Keywords

Main Subjects


[1] A. Ballester-Bolinches and L. M. Ezquerro, Classes of finite groups, Mathematics and Its Applications (Springer),
584, Springer, Dordrecht, 2006.
[2] W. Gasch¨utz, Praefrattinigruppen, Arch. Mat., 13 (1962) 418–426.
[3] W. M. Kantor and A. Lubotzky, The probability of generating a finite classical group, Geom. Ded., 36 (1990) 67–87.
[4] L. G Kov´acs and Hyo-Seob Sim, Generating finite soluble groups, Indag. Math. (N. S.), 2 (1991) 229–232.
[5] A. Lubotzky, The expected number of random elements to generate a finite group, J. Algebra, 257 (2002) 452–459.
[6] A. Lucchini, A bound on the expected number of random elements to generate a finite group all of whose Sylow
subgroups are d-generated, Arch. Math. (Basel), 107 (2016) 1–8.
[7] A. Lucchini, On groups with d-generator subgroups of coprime index, Comm. Algebra, 28 (2000) 1875–1880.
[8] A. Mann, Positively finitely generated groups, Forum Math., 8 (1996) 429–459. 
Volume 9, Issue 1 - Serial Number 1
Proceedings of the Ischia Group Theory 2018
March 2020
Pages 1-6
  • Receive Date: 14 August 2018
  • Revise Date: 13 November 2018
  • Accept Date: 20 November 2018
  • Published Online: 01 March 2020